A circular route to confine electrons

Physical barriers are used to confine waves. Whether it is harbor walls for sea waves, a glass disk for light, or the “whispering gallery” circular chamber walls in St. Paul's Cathedral for sound, the principle of confinement—reflection—is the same. Zhao et al. used that same principle to confine electrons in a nanoscale circular cavity in graphene. Periodic patterns within the cavity were associated with an electronic wave version of whispering gallery modes. The tunability of the cavity size may provide a route for the manipulation of electrons in graphene and similar materials.

Abstract

The design of high-finesse resonant cavities for electronic waves faces challenges due to short electron coherence lengths in solids. Complementing previous approaches to confine electronic waves by carefully positioned adatoms at clean metallic surfaces, we demonstrate an approach inspired by the peculiar acoustic phenomena in whispering galleries. Taking advantage of graphene’s gate-tunable light-like carriers, we create whispering-gallery mode (WGM) resonators defined by circular pn junctions, induced by a scanning tunneling probe. We can tune the resonator size and the carrier concentration under the probe in a back-gated graphene device over a wide range. The WGM-type confinement and associated resonances are a new addition to the quantum electron-optics toolbox, paving the way to develop electronic lenses and resonators.

Charge carriers in graphene exhibit light-like dispersion resembling that of electromagnetic waves. Similar to photons, electrons in graphene nanostructures propagate ballistically over micrometer distances, with the ballistic regime persisting up to room temperatures (1). This makes graphene an appealing platform for developing quantum electron optics, which aims at controlling electron waves in a fully coherent fashion. In particular, gate-tunable heterojunctions in graphene can be exploited to manipulate electron refraction and transmission in the same way that optical interfaces in mirrors and lenses are used to manipulate light (2). These properties have stimulated ideas in optics-inspired graphene electronics. First came Fabry-Pérot interferometers (3), which have been fabricated in planar npn heterostructures in single-layer graphene (4) and subsequently in bilayer (5) and trilayer graphene (6). The sharpness of the pn junctions achievable in graphene can enable precise focusing of electronic rays across the junction, allowing for electronic lensing and hyperlensing (7–9).

We report on electron whispering-gallery mode (WGM) resonators, an addition to the electron-optics toolbox. The WGM resonances are familiar for classical wave fields confined in an enclosed geometry—as happens, famously, in the whispering gallery of St. Paul’s Cathedral. The WGM resonators for electromagnetic fields are widely used in a vast array of applications requiring high-finesse optical cavities (10–12). Optical WGM resonators do not depend on movable mirrors and thus lend themselves well to designs with a high quality factor. This can render the WGM design advantageous over the Fabry-Pérot design, despite challenges in achieving tunability due to their monolithic (single-piece) character [see (12) for a mechanically tunable optical WGM resonator]. Our system is free from these limitations, representing a fully tunable WGM resonator in which the cavity radius can be varied over a wide range by adjusting gate potentials. In contrast, the best electronic resonators known to date—the nanometer-sized quantum corrals designed by carefully positioning adatoms atop a clean metallic surface (13)—are not easily reconfigurable.

Further, although WGM resonators are ubiquitous in optics and acoustics (10–12, 14), only a few realizations of such resonators were obtained in nonoptical and nonacoustic systems. These include WGM for neutrons (15), as well as for electrons in organic molecules (16). In our measurements a circular electron cavity is created beneath the tip of a scanning tunneling microscope (STM), and we directly observe the WGM-type confinement of electronic modes. The cavity is defined by a tip-induced circular pn junction ring, at which the reflection and refraction of electron waves are governed by Klein scattering (Fig. 1). Klein scattering originates from graphene’s linear energy dispersion and opposite group velocities for conduction and valence band carriers; Klein scattering at a pn junction features a strong angular dependence with a 100% probability for transmission at normal incidence, as well as focusing properties resembling negative refractive index metamaterials (2, 7). Although Klein scattering is characterized by perfect transmission and no reflection for normal incidence, it gives rise to nearly perfect reflection for oblique incidence occurring in the WGM regime (2). As illustrated in Fig. 1B, this yields excellent confinement and high-finesse WGM resonances for modes with high angular momentum m and a less perfect confinement for non-WGM modes with lower m values.

(A) The rings are induced by the STM tip voltage bias (Vb) and back-gate voltage (Vg), adjusted so as to reverse the carrier polarity beneath the tip relative to the ambient polarity. The pn junctions act as sharp boundaries giving rise to Klein scattering of electronic waves, producing mode confinement via the whispering-gallery mechanism. The cavity radius and the local carrier density are independently tunable by the voltages Vb and Vg. Electron resonances are mapped out by the STM spectroscopy measurements (see Fig. 2). Shown are the STM tip potential U(r) and the quantities discussed in the text: the STM tip radius (R), its distance from graphene (d), and the local (μ0) and ambient (μ∞) Fermi levels with respect to the Dirac point. n and p label the electron and hole regions. (B) Spatial profile of WGM resonances. Confinement results from interference of the incident and reflected waves at the pn rings (dashed lines). The confinement is stronger for the larger angular momentum m values, corresponding to more oblique wave incidence angles. This is illustrated for m = 5 (weak confinement) and m = 13 (strong confinement). Plotted is the quantity , the real part of the second spinor component in Eq. 1.

Electron optical effects in graphene have so far been explored using transport techniques, which lack spatial and angular resolution that would be indispensable for studying confined electronic states and/or electron lensing. Our scanning probe technique allows us to achieve nanometer-scale spatial resolution. The STM probe has a dual purpose: (i) creating a local pn junction ring, which serves as a confining potential for electronic states, and (ii) probing by electron tunneling the resonance states localized in this potential. The planar back gate and the STM tip, acting as a circular micrometer-sized top gate, can change both the overall background carrier density and the local carrier density under the tip. As such, pn and np circular junctions centered under the probe tip (Fig. 1A) can be tuned by means of the tip-sample bias Vb and the back-gate voltage Vg [see fig. S4 (17)]. For the purpose of creating resonant electronic modes inside the junction, this configuration gives us in situ, independent control over the carrier concentration beneath the STM tip and the pn ring radius. The tunneling spectral maps from such a device show a series of interference fringes as a function of the knobs (Vb,Vg) (Fig. 2). These fringes originate from resonant quasi-bound states inside the pn ring.

(A) Differential tunneling conductance (dI/dVb) for a single-layer graphene device, as a function of sample bias (Vb) and back-gate voltage (Vg). The gate map was obtained after increasing the probe-tip work function by exposure to deuterium to shift the interference fringes vertically downward (fig. S5) (17). The two fans of interference features, marked WGM′ and WGM′′, originate from WGM resonances in the DOS (see text). (B) Interference features in dI/dVb, calculated from the relativistic Dirac model. The features WGM′ and WGM′′ in the (Vg,Vb) map originate, respectively, from the conditions ε = μ0 and ε = μ0 + eVb (see text). The boundaries of the WGM′ (and WGM′′) regions are marked by dashed (and dotted) white lines, respectively. arb. units, arbitrary units. (C) dI/dVb spectra taken along the horizontal line in (A) at Vb = 230 mV. (D) dI/dVb spectra taken along the two vertical lines in the map in (A) at Vg = 16 V (red line) and Vg = –11 V (blue line, scaled ×3 and offset for clarity) (see text for discussion). The four peaks at positive bias at Vg = 16 V are fit to Gaussian functions, with the fits shown in the lower right of the figure. The peaks labeled 1′′,2′′,3′′… correspond to WGM resonances probed at the energy ε = μ0 + eVb, whereas the peaks labeled 1′,2′,3′…, are the same WGM resonances probed at the Fermi level ε = μ0, giving rise to the WGM′′ and WGM′ fringes in the gate maps, respectively. The resonance spacing of order 40 mV translates into a cavity radius of 50 nm, using the relation (see text).

The measured spacing between fringes (Δε) can be used to infer the cavity radius (r). Using the formula (ℏ, Planck’s constant h divided by 2π; vF ≈ 106 m/s) and an estimate from Fig. 2A (Δε ≈ 40 meV), we obtain r ≈ 50 nm, a value considerably smaller than the STM tip radius (R ≈ 1 μm). This behavior can be understood from a simple electrostatic model of a charged sphere proximal to the ground plane. When the sphere-to-plane distance d is small compared with the sphere radius R, the induced image charge density cloud ρ behaves as , predicting a length scale . This crude estimate is upheld, within an order of magnitude, by a more refined electrostatic modeling (17), which also gives a length scale much smaller than R.

The experimental results were obtained on a device consisting of a graphene layer on top of hexagonal boron nitride, stacked on SiO2 with a doped Si back gate [see supplementary materials for details (17)]. Figure 2A shows a tunneling conductance map as a function of back-gate voltage (Vg) on the horizontal axis and sample bias (Vb) on the vertical axis. A series of interference-like fringes forming a curved fan (labeled WGM′) can be seen in the upper right of Fig. 2A. The center of the fan defines the charge neutrality point. This point can be off (0,0) in the (Vg,Vb) plane due to impurity doping of graphene (shift along Vg) and the contact potential difference between the probe tip and graphene (shift in Vb). As illustrated in fig. S5 (17), we are able to shift the center point of the fan to lower Vb values by changing the tip work function, for example, with D2 adsorption (18). Another interesting feature in such conductance maps is a (somewhat less visible) second fan of fringes (labeled WGM′′), which is crossing the primary WGM′ fan. The fringes in the WGM′′ fan follow the typical graphene dispersion with respect to the Fermi energy, which varies with doping as from higher sample bias to lower as a function of Vg. Examining the primary (WGM′) and secondary (WGM′′) fringes more closely confirms that they originate from the same family of WGM resonances.

Figure 2C shows nine oscillations in a line cut across the WGM′ fan along the Vg axis. To understand the origin of these oscillations, we examine the two spectral line cuts along the Vb axis in Fig. 2D. The first spectrum in Fig. 2D at Vg = –11 V (blue curve) contains a group of resonances (labeled 1′′ to 3′′) near the Fermi level (Vb = 0) with a spacing of 37.6 ± 1.2 mV (19). In the map in Fig. 2A, these resonances can be seen to move to lower energies approximately following the typical Dirac point dispersion . Taking a vertical cut at a higher back-gate voltage of Vg = 16 V (red curve) shows resonances 1′′ and 2′′ shifted down in energy in Fig. 2D. Focusing now at slightly higher energies, the WGM′ resonances appear at positive energies in Fig. 2D and are labeled 1′ to 4′ for Vg = 16 V. These four resonances are fit to Gaussian functions and shown deconvolved from the background conductance in the bottom right of the figure. The average spacing of these resonances is 116.9 ± 7.5 mV (19). A close examination of Fig. 2A indicates the one-to-one correspondence between the WGM′′ resonances 1′′, 2′′,… and the WGM′ resonances 1′, 2′…, suggesting their common origin. We therefore conclude that the WGM′′ resonances correspond to tunneling into the pn junction modes at energy [μ0, local Fermi level (see Fig. 1A)], whereas the WGM′ resonances reflect the action of the STM tip as a top gate, allowing tunneling into the same resonance mode at ε = μ0 [see fig. S3 (17)]. For example, resonance 1′′ seen at Vb ≈ –100 mV is now accessible at the Fermi level by the tip-graphene potential difference, as shown in fig. S3D (17), when tunneling into the WGM′ resonance 1′ at Vb = 82 mV in Fig. 2A.

To clarify the WGM character of these resonances, we analyze graphene’s Dirac carriers in the presence of a potential induced by the STM tip described by the Hamiltonian , where H0 is the kinetic energy term and U(r) describes the STM tip potential seen by charge carriers. Because relevant length scales—the electron’s Fermi wavelength and the pn ring radius−are much greater than the atomic spacing, we focus on the low-energy states. We linearize the graphene electron spectrum near the K and K′ points, bringing H0 to the massless Dirac form: , where and are pseudospin Pauli matrices. We take the tip potential to be radially symmetric, reflecting the STM tip geometry. Furthermore, the distance from the tip to graphene (d) is considerably smaller than the electron’s Fermi wavelength and the pn ring radius, both of which are smaller than the STM tip radius. We can therefore use a parabola to approximate the tip potential, (r, off-center displacement). The curvature κ, which affects the energy spectrum of WGM resonances, can be tuned with the bias and gate potentials, as discussed in the supplementary materials (17).

The WGM states with different angular momentum can be described by the polar decomposition ansatz(1)where m is an integer angular momentum quantum number, is the polar angle, and A, B label the two graphene sublattices. We nondimensionalize the Schrödinger equation using the characteristic length and energy scales (, ) to obtain the radial eigenvalue equation of the two-component spinor u(r) with components uA(r) and uB(r)(2)Here r is in units of , is in units of , and is in units of . This equation is solved using a finite difference method [see supplementary material (17)]. We can use this microscopic framework to predict the measured spectral features. The tunneling current, expressed through the local density of states (DOS), is modeled as(3)which is valid for modest Vb values (20). Here μ0 is the Fermi energy under the tip, which in general is different from ambient Fermi energy μ∞ as a result of gating by the tip (see below). The transmission function T(ε,Vb), which depends on the tip geometry, work function, and DOS, will be taken as energy-independent. The quantity represents the sum of partial-m contributions to the total DOS beneath the tip, with ν labeling eigenstates of Eq. 2 with fixed m. The weighting factor is introduced to account for the finite size of the region where tunneling occurs, with the Gaussian halfwidth [see supplementary materials (17)].

The WGM resonances for different partial-m contributions , which combine into the total DOS (Fig. 3), reveal that individual WGM states exhibit very different behavior depending on the m value [see Figs. 1B and 3B]. Klein scattering at the circular pn junction produces confinement creating the WGMs, and the confinement is stronger for the large-m modes and weaker for small-m modes. The Klein reflection probability is strongly dependent on the angle of incidence θ at the pn interface, growing as , where ξ is a characteristic dimensionless parameter (21). The value of θ grows with m as . As a result, larger values of m must translate into larger reflectivity and stronger confinement. This trend is clearly demonstrated in Figs. 1B and 3B. Also, as m increases, mode wavefunctions are being pushed away from the origin, becoming more localized near the pn ring, in full accord with the WGM physics.

Fig. 3Contributions of the WGM resonances with different m to the DOS for the relativistic Dirac model.

(A) Colored curves represent partial-m contributions from angular momentum values m = 1,2,3,4,5 (see Eq. 3), evaluated for a confining potential with curvature value . Different curves show the partial DOS contributions defined in Eq. 3, which are offset vertically for clarity. The inset shows the total DOS versus particle energy ε and the curvature κ (see text). The black curve shows the total DOS trace along the white line. (B) The Dirac wavefunction for different WGM states (see Eq. 1). Spatial structure is shown for several resonances in the partial DOS (black dashed circles mark the pn junction rings). The quantity plotted, , is the same as in Fig. 1B. The length scale (the same in all panels) is marked. Note the confinement strength increasing with m.

To understand how one family of WGM resonances gives rise to two distinct fans of interference features seen in the data, we must account carefully for the gating effect of the STM tip. We start with recalling that conventional STM spectroscopy probes features at energies , where are system energy levels. This corresponds to the family WGM′′ in our measurements. However, as discussed above, the tip bias variation causes the Fermi level beneath the tip to move through system energy levels , producing an additional family of interference features (WGM′) described by [see fig. S3 and accompanying discussion (17)]. To model this effect, we evaluate the differential conductance G = dI/dVb from Eq. 3, taking into account the dependence μ0 versus Vb. This gives(4)with . The two contributions in Eq. 4 describe the WGM′ and WGM′′ families. We note that the second family originates from the small electron compressibility in graphene, resulting in a finite η and would not show up in a system with a vanishingly small η (e.g., in a metal). We use Eq. 4 with a value η = 1/2 to generate Fig. 2B. In doing so, we use electrostatic modeling [described in (17)] to relate the parameters (ε,κ) in the Hamiltonian, Eq. 2, and the experimental knobs (Vb, Vg). This procedure yields a very good agreement with the measured dI/dVb (Fig. 2, A and B).

In addition to explaining how the two sets of fringes, WGM′ and WGM′′, originate from the same family of WGM resonances, our model accounts for other key features in the data. In particular, it explains the large difference in the WGM′ and WGM′′ periodicities noted above. It also correctly predicts the regions where fringes occur (Fig. 2B). The bipolar regime in which pn junction rings and resonances in the DOS occur (see fig. S6 and S7) takes place for the probed energies ε of the same sign as the potential curvature. In the case of a parabola , this gives the condition εκ > 0, corresponding to the upper-right and lower-left quadrants in Fig. 3A, inset. However, under experimental conditions, the potential is bounded by (see Fig. 1A), which constraints the regions in which WGMs are observed (17). Accounting for the finite value yields the condition , with . This gives the WGM′ and WGM′′ regions in Fig. 2B bounded by white dashed and white dotted lines, respectively, and matching accurately the WGM′ and WGM′′ location in our measurements.

The range of m values that our measurement can probe depends on the specifics of the tunneling region at the STM tip. We believe that a wide range of m values can be accessed; however, we are currently unable to distinguish different partial-m contributions, because the corresponding resonances are well aligned (Fig. 3). Different m states may contribute if the tunneling center is not the same as the geometric center of the tip, which is highly likely. As shown in (13), higher m states can be accessed by going off center by as little as 1 nm, which is likely in our real experiment due to a residual asymmetry of the STM tip [we model this effect by a Gaussian factor in Eq. 3]. We note in this regard that different angular momentum m values translate into different orbital magnetic moment values, opening an opportunity to probe states with different m by applying a magnetic field.

The explanation of the observed resonances in terms of the whispering-gallery effect in circular pn rings acting as tunable electronic WGM resonators has other notable ramifications. First, it can shed light on puzzling observations of resonances in previous STM measurements (22–24), which hitherto remained unaddressed. Second, our highly tunable setup in which the electron wavelength and cavity radius are controlled independently lends itself well to directly probing other fundamental electron-optical phenomena, such as negative refractive index for electron waves, Veselago lensing (7), and Klein tunneling (2). Further, we envision probing more exotic phenomena such as the development of caustics, where an incident plane wave is focused at a cusp (25–27), and special bound states for integrable classes of dynamics, where the electron path never approaches the confining boundary at perpendicular incidence (28). These advances will be enabled by the distinct characteristics of graphene that allow for electronic states to be manipulated at the microscale with unprecedented precision and tunability, thus opening a wide vista of graphene-based quantum electron-optics.

Acknowledgments: Y.Z. acknowledges support under the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology Center for Nanoscale Science and Technology, grant 70NANB10H193, through the University of Maryland. J.W. acknowledges support from the Nation Research Council Fellowship. F.D.N. greatly appreciates support from the Swiss National Science Foundation under project numbers 148891 and 158468. L.S.L. acknowledges support from STC CIQM/NSF-1231319. We thank S. Blankenship and A. Band for their contributions to this project and M. Stiles and P. First for valuable discussions.